4 Climate Damages

We posit a damage process, \(N_t = \{ N_t : t\ge 0\}\) to capture negative externalities on society imposed by carbon emissions. The reciprocal of damages, \({\frac 1 {N_t}}\), diminishes the productive capacity of the economy because of the impact of climate change. We follow much of climate economics literature by presuming that the process \(N\) reflects, in part, the outcome of a damage function evaluated at the temperature anomaly process.

In our computational implementation, damage specification uses a piecewise log-quadratic specification as a function of the temperature anomaly \(y\). We suppose that the derivative of the logrithm of damages, \(\hat{n} = \log N\), with respect to temperature anomaly is

\[\begin{split}\begin{equation} \begin{array} {llll} \frac {d {\hat n}} {dy} & = & \lambda_1 + \lambda_2 y & y \le {\tilde y} \\ \frac {d {\hat n}} {d{\hat y}} & = & \lambda_1 + \lambda_2 {\hat y} + \lambda_3(\ell) ({\hat y} - {\bar y}) & {\hat y} > {\bar y} \end{array} \end{equation}\end{split}\]

where \({\hat y} = y + {\bar y} - {\tilde y}.\)

This equation has initial condition \({\hat n}(0) = 0,\) and we consider it for alternative possible values of \(\ell\) and \({\tilde y} \le {\bar y}.\)

In the stochastic version of what follows, \({\tilde y}\) will be triggered by a Poisson jump prior to a temperature threshold of \({\bar y}.\) We specify the intensity so that this jump takes place in the interval \([{\underline y}, {\bar y}].\) We shift the derivative of damages with respect to temperature to the right as captured by the move from \(y= {\tilde y}\) to \({\hat y} = {\bar y}\). We also increase the slope by including a term \(\lambda_3(\ell) ({\hat y} - {\overline y})= \lambda_3(\ell) ( y - {\tilde y})\) where the coefficient \(\lambda_3(\ell)\) is ex ante uncertain with a discrete distribution.

We plot the implied damage functions for thresholds between \({\tilde y} = 1.5\) and \({\tilde y} = 2.\), including a range of \(\lambda_3\)’s used in our quantitative policy assessment.

from src.plot import plot_damage
plot_damage("""Figure 2: Range of Possible Damage Functions for Different Jump Thresholds""")